Does a continuous state markov chain with stochastic transition density $K^n(x,y)>0$ for any $x$ and $y$ have a unique stationary distribution?

Question

I am trying to prove a discrete time continuous state markov chain has unique stationary distribution. The MC has the stochastic transition density $K^n(x,y)>0$ for every $x,y\in S$ and the state domain $S$ is an interval $[a, b]$ with $a$ and $b$ being finite. I am not familiar with MC with continuous state which is developed with a lot of measure theory. I guess I can prove this MC is irreducible and aperiodic but I am worried that these two conditions can not show the MC has a unique stationary distribution as null recurrency can also be an issue.

I was wondering whether you could give me some clue whether this MC indeed has this property and what are the relevant conditions to support a unique stationary distribution. I would appreciate if you could recommend me some literature which explains continuous MC intuitively. Thank you!

Answer 1

Alas I don't know "literature which explains continuous MC intuitively", but the following can be found in e.g. Durrett's Probability: Theory and Examples (4th edition. section 6.8). A general sufficient condition for uniqueness of, and convergence to, a stationary distribution is the minorization condition:

There exists a probability density $f$ and a constant $\delta > 0$ such that $K(x,y) \ge \delta f(y)$ for all $x,y$.

So in your case it is enough to show there exists $\varepsilon > 0$ such that $K(x,y) \ge \varepsilon $ for all $x,y$. This will follow from your stated assumption if the function $K(x,y)$ is continuous.